Ateneo - UPO

Rational mechanics, basic differential geometry, Lagrangian and Hamiltonian formalism.
Power series, series of functions, Fourier series. Fourier transform. Introduction to distributions.
Partial differential equations.

A. Fasano, S. Marmi, - Analytical mechanics Oxford, 2006

C. Bernardini, O. Ragnisco, P.M. Santini: Metodi Matematici della Fisica, La Nuova Italia Scientifica (A. Fasano, S. Marmi, Meccanica analitica : con elementi di meccanica statistica e dei continui, Bollati Boringhieri 2002)

N. Zanghì: Appunti di Metodi Matematici per la Fisica https://www.ge.infn.it/~zanghi/metodi/ZUL.pdf

M. Bramanti, C.D. Pagani, S. Salsa: Analisi matematica 2, Zanichelli

P. Olver, Introduction to Partial Differential Equations, Springer

To cover core mathematical methods in physics—including analytical mechanics—and train students to apply these mathematical tools correctly to physical problems

The mathematics courses of the previous year

Blackboard teaching, theory and exercise sessions. The blackboard teaching complements the textbooks: topics are presented developping and relating different key concepts, and underplaying learning by heart. Methods and concepts given as prerequisites are thus naturally recalled and are contextualized. This brings to a view on the new concepts and methods that is integrated and organic with the previous notions (ex. complex numbers and vectors). We favour an experimental approach to problem solving, based on trial and error. Checks on the correctness of results are encouraged and created by considering trivial subcases of the problems at hand. Homework exercises are proposed and highly recommended.

TIME REQUIRED The amount of time required to successfully pass the exam is highly dependent upon the soundness of the mathematical knowledge acquired during the previous years.
LEARNING CHECKS During every lesson the preceding one is recalled and summarized also through questions and sometimes the explanation of take home problems previously assigned. Students and teacher have therefore the possibility to assess the progresses in the learning process during the teaching of the course. See also the section: Modalita' di verifica dell'apprendimento. Students with physical disabilities, Learning Disabilities or Special Education Needs can request specific services and tools via the "Staff Sviluppo e Coordinamento Carriere e Servizi alle Studentesse e agli Studenti", and consulting the University webpage: https://uniupo.it/it/servizi/servizi-studentesse-e-studenti-condizione-di-disabilit%C3%A0-e-dsa . Students with disabilities, learning disabilities or special education needs, once they have contacted the University Staff, can refer to the tutor in charge of the course to define the examination modalities, concerning academic aspects.

Written exam. Oral exam right after. The written final exam lasts about two hours; a first part (6/30) of 6 questions (to be handed in after 30 minutes) is in order to check a minimum level of sudy and acquaintance with mathematical reasoning. Necessary condition to pass the exam is that of totalizing at least 4 out of the 6 points of this first part. The second part (24/30) is a set of 5-7 longer and more complex exercises. The oral part of the exam concerns the critical issues emerged in the written exam, and the logic analytic and deductive skills assessed by listening to the student presentation and the student ability to dialogue with the teacher. Excellence is reached with a written exam that has at most light calculations mistakes and misprints, and with an oral exams where questions are properly answered.

Analytical mechanics. Basic differential geometry. Lagrangian formalism. Symmetries and conservation laws. Hamiltonian formalism and canonical transformations. Function spaces. Fourier series. Fourier transform. Generalized functions and Dirac delta function. Fourier analysis of generalized functions. Applications to partial differential equations.

- Knowledge and understanding:
acquisition of the theoretical basis (theorems and definitions) of mathematical methods for applied sciences.
- Applying knowledge and understanding: Full ability to apply the computation techniques and analysis provided by the course.
- Communicative Skills: Being able to provide either written or oral details of the computation and of the results for the problem. Communicative skill regarding the computation procedure by means of a detailed analysis of computations.
- Learning skills: the student will have to acquire a certain mastery in the use of advanced mathematical tools to expand their knowledge for advanced courses.